No, actually I forgot the square in the numerator, since I was writing out the variance.
Still a stupid mistake on my part.

The variance is

[tex]
\frac{\sigma^2} n
[/tex]

so the standard deviation is

[tex]
\frac{\sigma}{\sqrt n}
[/tex]

Sorry...it appeared as if you were answering the OP's question about the standard deviation directly which is why I assumed you had finished the derivation all the way out to the standard deviation and not just the variance.